## 1 Introduction

[2] In the task of forecasting the atmospheric state, one can nowadays rely on different sources of information: forecasts provided by numerical models, field observations, and error statistics for both observations and model simulations. The availability of several numerical models reflects the fact that different physical parameterizations were derived to describe the same atmospheric phenomena. Also, the mathematical model could be turned into a numerical model by means of various numerical techniques. In addition, the input data can be provided by different sources and are usually uncertain to some extent. This suggests to consider an ensemble of forecasts: an ensemble brings together various sources of information and allows, therefore, to derive a new forecast which performs better than any individual ensemble member. The improved forecast is usually obtained by taking a linear or convex combination of ensemble members with some weights and is therefore called an aggregated forecast. Now, may be formulated as follows: construct weights such that the corresponding aggregated forecast is close in terms of some performance measure to the given reference which is usually represented by observations. The weights of the combination are updated as soon as new observations become available. So the procedure is referred to as sequential aggregation.

[3] One of the main problems of forecasting algorithms based on ensemble aggregation is uncertainty estimation. Given various numerical models and related observations, with different uncertainty descriptions, one needs to combine these descriptions all together and transform them into an estimate of the uncertainty associated with the aggregated forecast. We stress that the weights of the aggregated forecast change when new observations become available; hence they evolve over time. Thus, the uncertainty transformation should be accomplished by the aggregation algorithm along with the evolution of the weights. In other words, the dynamics of the weights drives the aggregated forecast and its uncertainty estimate. Since this dynamics is uncertain, one needs to assume an appropriate uncertainty description for the weights. We stress that technically the error statistics for observations may be incomparable with uncertainty description for the weight's evolution or even for individual ensemble members: for instance, the measurement error may be stochastic, and the error of the numerical models may be deterministic. The minimax filter can handle such case and provide an uncertainty estimation for the weights (hence for the aggregated forecast as well) in the form of a bounding set.

[4] Another important technical issue to consider is the sparsity (in space) of the observation's network. It may result in weights that are optimal only locally, i.e., at observed locations. This problem is solved using “ensemble forecast of analyses” (EFA) [*Mallet*, 2010]. In EFA, one first uses a data assimilation algorithm to generate an analysis. At a given date and under given assumptions, the analysis is the a posteriori estimate of the atmospheric state that optimally combines (in the least squares sense) observations, simulations, and error statistics. The main idea of EFA is to forecast forthcoming analyses instead of observations. The analyses are the preferred target because they include all a posteriori knowledge on the atmospheric state, they take into account observational errors, and they provide comprehensive information (i.e., concentrations for all pollutants in all model grid cells). The weights of the aggregated forecast are thus adapted to forecast the analyses instead of the observations, and the weights can depend on space (one weight per model and per state component).

[5] In air quality applications, the aggregation of ensemble simulations has been carried out by different strategies. *Delle Monache and Stull* [2003] relied on the ensemble mean, where all simulations were given the same weight. Analysis of ensemble mean variance was carried out in *Potempski and Galmarini* [2009] and *Solazzo et al.* [2012]. Multimodel forecast based on ensemble median was reported in *Riccio et al.* [2007]. Bias correction techniques were tested in *Monteiro et al.* [2013]. Nonstationary weighting procedure for ozone forecasting based on a dynamic linear regression was presented in *Pagowski et al.* [2006]. Our approach also assumes an equation for the dynamics of the aggregation weights and an observation equation. Note that the latter, in fact, allows one to compare observations (or analyses) and aggregated forecasts corresponding to given weights. However, our uncertainty description differs. In the dynamic linear regression, the errors—in the weights equation and observation equation—are Gaussian and the variance of the observational error is unknown. In contrast, the minimax approach assumes that uncertainty description is given in terms of bounding sets: the errors in the weight equation are elements of a prescribed set. Similarly, the variances of the observational errors may not be prescribed, but they must belong to the given set as well. As a result, the algorithm is robust with respect to uncertainty in observation error covariance matrix unlike Kalman filter: it is well known from the control literature (see, for instance, *Shen and Deng* [1997] and *Başar and Bernhard* [1995]), that *H*_{2} or Kalman state estimators may be sensitive to uncertainty in the statistical error description. In other words, small perturbations in error covariance matrices may lead to significant deviations in estimates and/or error estimates. Such perturbations are often introduced in practice because the covariance parameters are usually estimated from data. *Mallet and Sportisse* [2006] used plain least squares methods on a large ensemble. *Mallet et al.* [2009] applied several machine learning algorithms on the same ensemble, especially a version of the ridge regression with discount in time. Machine learning algorithms are robust, adapted to operational forecasting and guarantee good performance in the long term. However, contrary to dynamic linear regression or minimax approach, they do not evaluate the uncertainty associated with their forecasts. Also, we prove that the weights obtained by means of the discounted ridge regression can be generated by our algorithm for a suitable choice of parameters.

[6] The main contribution of this paper is a minimax aggregation algorithm and uncertainty estimate associated with forecasts together with a simple method to check its reliability. Our aggregation method is based upon a minimax state estimation approach [see *Bertsekas and Rhodes*, 1971; *Nakonechny*, 1978; *Milanese and Tempo*, 1985; *Milanese and Vicino*, 1991; *Kurzhanski and Vályi*, 1997]. The estimation problem is defined as follows: given a state equation (the model, for the weights), an observation equation, and error descriptions, one needs to estimate the state of the model assuming that model errors belong to a given bounding set, and observational errors are realizations of random variables with given mean and unknown variance—which is in a given bounded set as well. To solve the estimation problem, we construct a worst-case error which selects the worst possible realization of uncertain parameters and leads to the maximal possible estimation error. The minimax estimate is chosen to have the least possible worst-case estimation error. This, in turn, allows us to construct a set of all possible estimates that are compatible with the model, observed data, and uncertainty description. This set represents, in fact, an uncertainty estimate.

[7] In this work, we construct the minimax estimate for aggregation weights in the form of a linear recursive filter. The so-called state equation (or model) defines the weights dynamics. The so-called observation equation actually involves the analysis, which is supposed to be equal to a linear combination of the ensemble of simulations, plus some unknown error. The filter estimates the aggregation weights which are in turn used to compute an aggregated forecast. The filter also provides an uncertainty estimation for the weights, from which the forecast uncertainty can be derived. In addition, we propose a method to check the reliability of the uncertainty estimation. We stress that the bounding set for the initial weight may be unbounded reflecting the fact that the initial weight may be chosen arbitrarily. This, in turn, proves that the algorithm's convergence does not depend upon a choice of the initial weight. In addition, the algorithm is robust to the variation in observation error covariance matrices unlike Kalman filters as it was mentioned above.

[8] The paper is organized as follows. Section 2 introduces a version of the minimax filter dedicated to sequential aggregation. It explains how to compute and assess the uncertainty estimation and discusses the links to Kalman filter and discounted ridge regression. Section 3 introduces the application to air quality, with further explanations on the EFA strategy. It briefly describes the ensemble simulations and the generation of the analyses. It also gives the parameters related to the minimax filter. Section 4 evaluates the forecast performance and uncertainty estimations. We also address the sensitivity of the results.